Step 1: Understand why a cell stops producing current.
An electrochemical cell works as long as redox reaction can proceed and reactants are present.
Step 2: What happens with time.
At the anode, oxidation occurs and metal dissolves into solution as ions, so the anode electrode gradually gets consumed.
Similarly, reactants in electrolyte can also get used up.
Step 3: When electrode is consumed.
If one electrode is completely eaten away, the redox process cannot continue, so the cell stops working.
Step 4: Choose correct option.
Thus option (C) is correct.
Final Answer:
\[
\boxed{\text{(C) one of the electrodes is eaten away}}
\]